The present invention relates to a method to evaluate and estimate relative permeability for fractional multi-phase, multi-component fluid flow through porous media. Relative permeability data estimated with the present method can be used, for example, in many areas such as oil field simulation, estimating oil or gas production rates, estimating recoverable reserves, designing hydrocarbon recovery strategies such as fracturing or “fracking,” life sciences, paper manufacturing, food industry, agriculture, and other areas relating to geology and geophysics. The present invention also relates to a computerized system and components thereof for performing such a method.
Relative permeability (“kr”) is used to quantify multiphase flow, such as the flow of oil in the presence of water and water in the presence of oil. In a sample with two such fluids, the relative permeabilities km and krw, by definition, are given by equations [9] and [10]:
                              k          m                =                  -                                                    Q                n                            ⁢                              μ                n                                                                    k                Absolute                            ⁢                                                A                  ⁢                                      ⅆ                    p                                                                    ⅆ                  x                                                                                        [        9        ]                                          k          rw                =                  -                                                    Q                w                            ⁢                              μ                w                                                                    k                Absolute                            ⁢                                                A                  ⁢                                      ⅆ                    p                                                                    ⅆ                  x                                                                                        [        10        ]            where the subscripts “n” and “w” refer to non-wetting fluid and wetting fluid, respectively. The fluxes Qn and Qw are measured at fixed saturation Sw. Relative permeability is usually plotted versus Sw.
The relative permeability depends on more factors than kAbsolute, including the wettability of the fluids and minerals system, interfacial surface tension, and viscosity contrast between the fluid phases, the velocities of the fluids, the saturation level of the fluid in the pores, the structure and connectivity of the pores in the porous solid and the pore space geometry. Another important factor that influences the relative permeability is the time history of the flows that went through the porous media. These parameters may vary in space and time and the resulting fluid state and composition changes during production of fluids.
In a porous medium, capillary attraction is determined by the adhesion between a liquid present in the body and the body itself and by the cohesive force of the liquid to itself. A liquid that wets a solid surface has greater adhesion to the particular solid than a non-wetting fluid. A fluid may wet one solid and not another solid. In multiphase fluid flow, wettability is a relative property. For example, if the force of adhesion of a first fluid for a porous medium is greater than the force of adhesion of a second fluid for a porous medium, then the first fluid is said to be wetting and the second fluid is said to be non-wetting.
Saturation, Sx, is the volume fraction of the total pore volume in a porous medium that is occupied by material “X”. The saturation level is a value between 0 and 1. A saturation level of 1 indicates that the entire available pore space in a given porous medium is filled by the fluid under consideration. Relative permeabilities are a function of fluid saturation. As the saturation of a particular phase increases, its relative permeability increases. Saturation history also has a major effect on relative permeability. The relative permeability-saturation relationship exhibits a hysteresis effect between the drainage process (wetting phase decreasing) and imbibition process (non-wetting phase increasing). It is believed that most subterranean porous rock formations were initially water filled and hydrocarbons entered these porous formations displacing part of the water. This history must be reproduced or assessed before any estimation of relative permeability is attempted so that realistic starting conditions are established. Imbibition and drainage plots of relative permeability versus saturation are shown in FIG. 1.
When a porous medium contains two or more immiscible fluids, the local volume of material in any particular pore may be different from the overall or average saturation level for the entire porous rock sample. For example, one fluid may strongly adhere to the surfaces within a given pore while another material may have no effective contact with the solid material. The local pore space geometry within a given porous medium can vary considerably and these variations in geometry can affect local saturation levels.
In practice, relative permeability can be estimated by physical lab tests or by numerical simulations.
One of the early physical lab methods for measuring relative permeability is described in U.S. Pat. No. 2,345,935 (Hassler). The method involves sealing all but two opposing surfaces on a porous rock sample. A fluid or fluids under pressure are introduced into one open surface and forced to flow through the sample at a specified flow rate. Fluid pressures are generated by pumps or similar means. The pressures and flux rates are inputs to the relative permeability calculation. One shortcoming of the Hassler technique is the need to determine internal wetting fluid pressures within the porous medium. This problem is described by W. Rose, “Some Problems in Applying the Hassler Relative Permeability Method,” 32 J. Petroleum Technology, 1161-63 (July, 1980). U.S. Pat. No. 4,506,542 (Rose) describes an apparatus and method that does not require measurement of internal pressures for estimation of relative permeability.
The Hassler method is a Steady State Method that can be used to calculate relative permeability versus saturation for a full range of saturations from 0 to 1. For two phase systems of immiscible fluids, the rock sample may first be purged with one fluid for a sufficient time such that the saturation in the rock sample of the selected fluid is 1. Then the other fluid or combinations of the two fluids are forced through the sample for a sufficient time to achieve steady state of the two fluxes Qn and Qw. At this point, the flux and pressure readings can be used to calculate kn, kw, for a given value of Sw, and plotted. The ratio of wetting and non-wetting fluids at the inlet of the sample can then be changed. This new combination of wetting and non-wetting fluids are forced through the sample for a sufficient time to achieve steady state of the two fluxes Qn and Qw. Another pair of relative permeabilities, kn, kw corresponding to another value of Sw, are calculated and another point is plotted. By repeating this procedure for different combinations of wetting and non-wetting fluids, a graph of relative permeability versus saturation can be plotted as shown in FIG. 2.
Other steady state physical methods to compute relative permeability include the Penn State Method (Snell, R. W., Measurements of gas-phase saturation in a porous medium, J. Inst. Pet., 45 (428), 80, 1959; The Hafford method (Naar, J. et al., Three-phase imbibition relative permeability, Soc. Pet. Eng. J., 12, 254, 1961); the Single-Sample Dynamic Method (Saraf, D. N. et al., Three-phase relative permeability measurement using a nuclear magnetic resonance technique for estimating fluid saturations, Soc. Pet. Eng. J., 9, 235, 1967); the Stationary Fluid Method (Saraf, D. N. et al., Three-phase relative permeability measurement using a nuclear magnetic resonance technique for estimating fluid saturations, Soc. Pet. Eng. J., 9, 235, 1967); and the Dispersed Feed Method (Saraf, D. N. et al., Three-phase relative permeability measurement using a nuclear magnetic resonance technique for estimating fluid saturations, Soc. Pet. Eng. J., 9, 235, 1967).
Another method, the Un-Steady State Method, also begins with the rock sample initially saturated with the wetting fluid. Then the non-wetting fluid is forced through the sample, the fraction of non-wetting fluid recovered and the pressure drop across the sample are recorded and used to calculate various combinations of kn, kw at corresponding values of Sw.
Laboratory methods can suffer from a number of shortcomings, which may include one or more the following:                1. The sample to be tested is in the lab at surface conditions whereas the in-situ sample may be at temperatures above 100° C. and 100-700 bar. When samples are brought to the surface many properties of the rock change. Creating artificial conditions to replicate downhole conditions is difficult, expensive, and/or imprecise.        2. The pressures required to achieve desired flow rates may be extremely high causing leakage problems and/or equipment malfunctions.        3. A large volume of fluid must be processed for the sample to come close to steady state.        4. Tests can take a very long time up to weeks or months or more than a year to complete.        5. Very tight formations such as shales may be difficult or impossible to measure.        6. Initial conditions such as saturation, wettability, and fluid distributions are difficult to establish.        7. Establishing wettability in the lab is difficult because cores are usually cleaned prior to the testing and initial wettability cannot be accurately restored.        8. In the lab, it is difficult and expensive to conduct tests with reservoir fluids at reservoir conditions. Mixing gas and oil at reservoir temperatures and pressures is difficult and can be dangerous.        
Numerical simulations to calculate relative permeability typically use numerical methods such as pore network modeling or direct simulation of multi-phase/multi-component flow in a porous medium.
One such general method to compute relative permeability is described in U.S. Pat. No. 6,516,080 (Nur). This method as with most numerical methods relies on production of a digital representation of a porous medium, hereinafter referred to as a “Sample,” for which relative permeability is to be estimated. The digital representation is typically produced by a CT X-ray scanner and then refined to compensate for limitations in resolution of the scanner. This representation along with fluid properties, rock properties, initial saturation, wettability, interfacial tension and viscosities are used as input to the lattice Boltzmann algorithm. The Lattice-Boltzmann method is a tool for flow simulation, particularly in media with complex pore geometry. See, for example, Ladd, Numerical Simulations of Particulate Suspensions via a discretized Boltzmann Equation, Part 1: Theoretical Foundation, J. Fluid Mech., v 271, 1994, pp. 285-309; Gunstensen et al., “Lattice Boltzmann Model of Immiscible Fluids, Phys. Rev. A., v. 43, no. 8, Apr. 15, 1991, pp. 4320-4327; Olsen et al., Two-fluid Flow in Sedimentary Rock: Simulation, Transport and Complexity, J. Fluid Mechanics, Vol. 341, 1997, pp. 343-370; and Gustensen et al., Lattice-Boltzmann Studies of Immiscible Two-Phase Flow Through Porous Media,” J. of Geophysical Research, V. 98, No. B 4, Apr. 10, 1993, pp. 6431-6441). The Lattice-Boltzmann method simulates fluid motion as collisions of imaginary particles, which are much larger than actual fluid molecules, but wherein such particles show almost the same behavior at a macroscopic scale. The algorithm used in the Lattice-Boltzmann method repeats collisions of these imaginary particles, and provides a distribution of local mass flux.
The accuracy of numerical methods to calculate relative permeability such as the Nur method depends in part on the accuracy of the Sample. The Sample is made up of discrete elements called voxels. Voxels are volumetric pixels. A digital representation of a three-dimensional object can be sub-divided into voxels. Ideally, each voxel is accurately classified as either solid or pore. The choice between solid or pore may not always be clear due to differences in the resolution of the scan and the minimum size of the grains in the porous medium. If a voxel is classified as solid, the nature or composition of the solid also should be known or determined in order to numerically model and make estimates of its physical properties.
In addition, the accuracy of numerical methods to compute relative permeability also depends on the numerical methods applied. The robustness of these methods can depend upon how boundary conditions in the algorithm are handled. There can be inlet and outlet boundary conditions, boundary conditions on the top, bottom, left or right of the sample and boundary conditions on the interior of the porous medium. The latter include effects on wettability especially when relatively small fractional flows of one fluid or the other are present. Boundary conditions are a quite complex problem in numerical methods. Selection of boundary conditions can significantly affect the time required for computation, the accuracy of results and the stability of the simulation. This can be especially true for immiscible multi-phase or multi-component simulations. Difficulties can arise from the fact that the pressure and distribution of the phases and velocities at the inlet of the digital simulation are unknown and these conditions must be established such that they mimic the physical conditions. There is no standardized and unique way of setting appropriate boundary conditions and many authors propose their own solution. The boundary conditions chosen can be of primary importance since they significantly can affect the numerical accuracy of the simulation and also its stability.
Numerical methods can have advantages over laboratory methods, such as in one or more of the following ways.
1. Because numerical simulations are virtual, they do not require the physical presence such as downhole fluids at downhole conditions. In the case of relative permeability in oil and gas formations, hydrocarbons at high temperatures and pressures, often above the critical point, are difficult to control and dangerous to handle.
2. Because numerical simulations can accelerate the time scale used, numerical simulations can be completed in a matter of hours or days instead of weeks, months, or longer. Because of this, more variations in fluid composition and flux can be processed using numerical methods than are practical in lab tests.
3. Numerical simulations have the advantage that the properties of any component can be accurately calculated at any location and at any time.
Numerical methods also may suffer some drawbacks, including one or more of the following:                1. Initial and boundary conditions are difficult or impossible to assess which results in inability in some cases to accurately calculate relative permeabilities or instability in computation. This is especially true when fractional flow of one or more components is small.        2. The distribution of wettability in space and time within a porous medium is difficult to assess.        
The present investigator has recognized that there is a need for new methods and systems for simulating fluid flow through porous media to provide, for example, improved evaluations and estimates of relative permeability for fractional multi-phase, multi-component fluid flow through the porous media, and which can provide an improved capability to evaluate and estimate the potential productivity of an oil field or other subterranean reservoir, and/or which may provide improved modeled estimates of fluid flow through other types of porous media or other uses.